1918]
CLOUD OF SIMILAR SMALL PARTICLES OF ANY SHAPE
545
and, finally, the averaging with respect to Q gives
T mean/ =
9 AE - - —
(19)
This represents the intensity of the vibrations parallel to X dispersed along OF, due to primary vibrations parallel to Z. It vanishes, of course, if A = B = C; while, if A = B merely, it reduces to (4). The ratio of the two polarized components is
A* + B* + C*-AB-BC- OA ,
3(A* + B2 + C*) + 2(AB + BC+CA)' ............... ( '
reducing to (5) when B = A.
If the primary light travelling in direction OX is unpolarized, we have also to include primary vibrations parallel to Y. The secondary vibrations scattered along OY are of the same intensity whether they are parallel to Z or to X. They are given by (19), where all that is essential is the perpendicularity of the primary and secondary vibrations. Thus, in order to obtain the effect along OF of unpolarized primary light travelling • along OX, we have merely to add (19) to both components. The intensity of the component vibrating parallel to Z is thus
ft {3 (A* + B* + C") + 2 (AB + EG + OA)}
.(21) .(22)
.(23)
while that of the component vibrating parallel to X is simply
^{42 + J?2+(72- AB-BC-CA] The ratio of the two intensities is .
reducing to (6) when B = A.
It may be observed that, since (21) = (14) 4-(19), we obtain the same intensity whether we use a polarizer transmitting vibratiQns parallel to Z and no analyser, or whether we use an analyser transmitting vibrations parallel to Z and no polarizer.
If neither polarizing nor analysing apparatus is.employed, we may add (21) and (22), thus obtaining
ft [6 (A2 + B2+C2)- AB-BC-CA]................(24)
When the particles are supposed to be of uniform quality, with a specific inductive capacity K' as compared with K for the undisturbed medium, and to be of ellipsoidal form with semi-axes a, b, c, we have
•tr>_ jr rrt__ rr rrt_ rr
A —1 • B~l • C*~l = 1 4- T. • "I _i_ _r M". 1 i AT f 25")
4nrK ' 4nrK ' ^nrK ' ......
R. vi. 35separately vibrations parallel to Z and to X. As regards the former, we have the same set of factors over again, as in (1), so that the vibration is A sin2 6 + C cos2 6, reducing to G simply, if A = G. This is the result for a single particle whose axis is at W. What we are aiming